21 PRODUCT OF MEASURES 231 (b) Show that for / to be ^t-integrable it is necessary and sufficient that D/ should be v-integrable, and that we have then v(D/) = /'dp. Moreover, if g is the decreasing real-valued function on R+ defined by g(t) = p*(f~l [t, +00]), then (fdjjL = [** g(t)dt. (c) Suppose that X is an interval [0, a] (where a > 0) and take ft to be Lebesgue measure on X. If/is a decreasing /x-integrable function, show that /• /^<A> f(t)dt<, f(t)dt JA Jo for every /x-measurable subset A of X. (Use (b).) L (a) With the notation of Problem 1, let Tf be the graph of/, i.e., the set of all points (#,/(*)) in X X R. If/is /z-measurable, show that Pf is v-negligible (use (13.21.13) and (13.21.10)). (b) Let 0 < a < 1 and let Q(a) be the subset of R2 which is the complement of the union of ]-l,l[x]— a,a[and] — a, a[x ] —1,1 [in the squareQ = [-1,1] X [-1,1]. The set Q(a) is the union of its four connected components Ch(a) = [-1, -a] X [-1, -a], Q2(a) = [-1, -a] X [a, 1], Q3(a)= [a, 1] X [a, 1], Q4(a) = [a, 1] X [-1, -a]. For i = 1, 2, 3,4 let hit a denote the similitude mapping Q to Qi(a) for which /h,a(-l, -!) = (-!, -1), Alf.(l, -!) = (-!, -a), h2, «(-!, -0 = (-1, a), h2t .(1, -1) - (-a, a), h3.«(-!, -1) = («, a), h3,8(1, -1) = (1, a), /U. .(-1, ™ D = (1, -a), V a(l, -1) = (1, -1). Also, in the interval [0, 7] in R, put Ifc = [A, k -f 1 ] for 0 g k <; 6 and let w* be the increasing similitude which maps [0, 7] to I*. Let/a be the continuous mapping of [0, 7] into Q which is affine-lineaf on each of the intervals I* and is such that the images of 0, 1, 2,..., 7 are respectively the points (-1, -1), (-1, -a), (-1, a), (-a, a), (a, a), (1, a), (1, -a), (1, -1). We shall now define a sequence (gn) of continuous mappings of [0, 7] into Q as follows. Let (an)n>0 be a decreasing sequence of numbers belonging to the interval ]0,1 [. Define g0 ==/«0. For n S> 1, suppose that gn~i has been defined, and consider all sequences s = (j\,..., /„) in which each 4 is one of the integers 0, 1,..., 6. Put t;s = Ull o • • - Q uin. Then it is sufficient to define gn(vs(t)) for 0 ^ / <; 7 and for each of the 7" sequences s. If at least one of the /i is odd, we put ^(^(O) = #n-i(us(0)- If on the other hand /t = 2jt for 1 g / <i n (with 0 <>ji g 3), we put#w(t>s(/))« w,(/ai>(0), where H>S = hJl +1, ai ° hj2 + it a2 <= • • • o hjn+i( «„. Show that the sequence (gn) converges uniformly to an injective continuous mapping g : [0, 7] -> R2, and that the simple arc g([Q, 7)) (Section 4, Appendix to Chapter IX) is nonnegligible with respect to Lebesgue measure on R2, if the sequence (an) is suitably chosen. L (a) Give an example of two compact spaces X and Y, positive measures A and fju on X and Y, respectively, and a A ® ^-measurable function / such that the two integrals djj,(y) I /(;c, y) d\(x) and I dX(x) f(x, y) d^(y) are both defined and are unequal (cf. Section 5.2. Problem 5).b) Let x0eX. For the series £(#|/,)/n(#o) to converge for every function